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The intensive pulse sound wave can be generated by the underwater plasma sound source (UPSS) based on the discharge of the underwater high voltage. The distribution of the sound field is prominently nonlinear. In this paper, the sound field of the intensive UPSS is described by the integral two-dimensional axisymmetric unsteady Euler equations firstly. In order to solve the Euler equations numerically, an optimized fifth-order symmetric WENO (weighted essentially nonoscillatory) method based on the three templates is proposed which is called WENO-SYM3. Without increasing the number of candidate templates, a new symmetric template structure can be obtained by expanding the second template and shifting the third one backwards for one space. The method is validated through numerical examples and experiments, and the results show that WENO-SYM3 has a high distinguished accuracy; meanwhile, its nonphysical oscillations are not obvious. The experimental results are basically the same as the calculation results, and the maximum error is around 3%.

After the development in recent half a century, the technology of high intense pulse sound wave, generated by the discharge of underwater plasma based on the liquid-electric effect [

The sound field distribution of a large amplitude pulse sound wave generated by UPSS is prominently nonlinear. Usually, the Schlieren method of photography is used to observe the propagation of intense sound field by the means of high-speed camera or pressure sensors which are used to measure the sound pressure of the sound field [

In this paper, an optimized fifth-order symmetric WENO (weighted essentially nonoscillatory) method (WENO-SYM3) based on the three templates is proposed [

Generally, the sound field modeling of the underwater plasma intense sound adopts Euler equations. The integral Euler equations can be expressed as

Since there is no analytic solution of the Euler equations, numerical method can be used to solve it. As a high-resolution numerical method, WENO has been more and more widely used to solve the Euler equations. The alternative template of the normal WENO scheme [

Sketches of two kinds of WENO schemes.

To improve the performance of the WENO scheme, Weirs [

Flow field parameters can be stored in the center of the control volume if the cell-centered method is chosen. According to the sum of flux density on the control volume boundary, the derivation of the shift can be calculated. By considering the following examples focusing on the right side of the control volume boundary, on which the flow field parameter is

Assuming that the initial template is

The definitions of these coefficients

Optimized symmetry of WENO-SYMOO scheme is realized by adding a candidate template. In this section, a novel optimized fifth-order symmetric WENO method based on the three templates (WENO-SYM3) is proposed, which easily realizes optimized symmetry of the algorithm (as shown in Figure _{1} without changing the number of candidate template.

Sketch of the optimized symmetrical WENO scheme.

The smoothness measurement coefficient

The coefficients

Finally,

After the flow field parameters of the control volume boundary are calculated, this problem could be translated into solving the Riemann problem, and this method was firstly proposed by Godunov [

Flow field parameters, such as the flow density

As for the time discretization, the TVD Runge-Kutta method with three order accuracy is more popular [

In order to validate whether the numerical method is correct or not, we need to compare the numerical result with the exact solution and to find some difficult numerical examples (close to real conditions) to test the effectiveness of the performance of the numerical method. It is more helpful to show the superiority of this algorithm. The correctness and effectiveness of numerical method are validated by Lax shock tube problem and Shu-Osher problem. For the first one, the exact solution existed. Shu-Osher problem contains the fine structure of shock wave and smooth flow field, which is a simple model of shock wave and turbulence effect. The effectiveness and computational accuracy can be tested by this example.

The mathematical model of Lax shock tube problem is as follows:

The initial state is

WENO-JS method, WENO-SYMOO method, WENO-SYM3 method, and MUSCL method are used to solve this problem in simulation with grid number 400, CFL = 0.1, and total time

Density solution to the one-dimensional shock tube problem for the various schemes.

The comparison of computation time before and after optimization.

Grid number | WENO-SYMOO | WENO-SYM3 | Save time |
---|---|---|---|

100 | 5.88 | 5.40 | 8.16% |

200 | 19.26 | 18.21 | 5.45% |

400 | 69.55 | 66.34 | 4.62% |

800 | 267.65 | 254.87 | 4.77% |

In Figure

The initial conditions of one-dimensional Euler (

Computational domain is [−5,10]. This example is a model that contains the fine structure of shock wave and smooth flow field, which is a simple model of shock wave and turbulence effect. So shock wave and small flow structure appear together as it contains dextral positive shock wave entering flow field with density fluctuate. There are high requirements for resolution of format and diffusion of value in numerical simulation. In this paper, speeds in different positions are given when grid number is 300, CFL = 0.1, and

Velocity solution to Shu-Osher’s problem by the various schemes.

For one-dimensional shock tube problem, there is a precision difference in the results of the two methods, but it is not too big. As shown in Figure

In consideration of the features of UPSS studied in this paper, namely, high pressure peak and narrow pulse width, we choose WENO-SYM3 method as numerical calculation method to calculate propagation characteristics of UPSS field by comparison of the two examples above, and certain related experiments were designed to validate the accuracy of the results.

A schematic diagram of the experimental setup is shown in Figure

Schematic diagram of the experimental setup.

The function of power-storage capacitor C is to store and discharge the voltage power. The charge-discharge control system would control the trigger circuit to generate trigger pulse, which would break down and turn on the spark gap switch G. Once the spark gap switch G is triggered, the discharge electrode S will be broken down. Then, a mass of electrical power is dissipated into the discharge channel. As a consequence, the sound wave is generated and measured at the main axis by pressure sensors. In the same time, the data is recorded by using a multichannel oscilloscope.

Due to the symmetry of UPSS propagates in open water, we choose a quarter of physical region as the calculating region, in which

Initial condition.

The nephograms of the sound field calculated at different times are shown in Figure

Sound field at different times.

The attenuation rule of pulse sound propagation in

Attenuation rule of the sound.

In order to validate the correctness of simulation results, related experiment was designed. Experiment was carried out in a water tank of 3 × 1 × 1 m. The capacitance of capacitor was 5

Comparison of the experimental with the simulation wave.

The capacitance of capacitor was chosen as 5

Experimental data at different locations with different discharge voltages.

In the experiment, the pressure at the sound source varied with the discharge voltage. Thus, the simulation of different discharge voltages could be realized by setting different initial pressures in simulation. The simulation results and experimental data of direct wave amplitude at the distances of 0.24 m, 0.48 m, and 0.877 m were listed in Figure

Comparison of the experimental and simulation results.

An optimized fifth-order symmetric WENO method (WENO-SYM3) is proposed based on the two-dimensional axisymmetric unsteady Euler equations in order to calculate UPSS sound field. Compared with the common WENO method and symmetry method in [

In the last of the paper, UPSS sound field was calculated by WENO-SYM3 method, and related experiment was conducted to validate the results of calculation. The results show that calculation results are consistent with experimental data with the maximum error about 3%.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (61701529, 61471298) and the Science and Technology on Electronic Information Control Laboratory Foundation of China (N2016KC0029).